Neural Computing and Applications

, Volume 27, Issue 4, pp 1053–1073 | Cite as

Dragonfly algorithm: a new meta-heuristic optimization technique for solving single-objective, discrete, and multi-objective problems

Original Article

Abstract

A novel swarm intelligence optimization technique is proposed called dragonfly algorithm (DA). The main inspiration of the DA algorithm originates from the static and dynamic swarming behaviours of dragonflies in nature. Two essential phases of optimization, exploration and exploitation, are designed by modelling the social interaction of dragonflies in navigating, searching for foods, and avoiding enemies when swarming dynamically or statistically. The paper also considers the proposal of binary and multi-objective versions of DA called binary DA (BDA) and multi-objective DA (MODA), respectively. The proposed algorithms are benchmarked by several mathematical test functions and one real case study qualitatively and quantitatively. The results of DA and BDA prove that the proposed algorithms are able to improve the initial random population for a given problem, converge towards the global optimum, and provide very competitive results compared to other well-known algorithms in the literature. The results of MODA also show that this algorithm tends to find very accurate approximations of Pareto optimal solutions with high uniform distribution for multi-objective problems. The set of designs obtained for the submarine propeller design problem demonstrate the merits of MODA in solving challenging real problems with unknown true Pareto optimal front as well. Note that the source codes of the DA, BDA, and MODA algorithms are publicly available at http://www.alimirjalili.com/DA.html.

Keywords

Optimization Multi-objective optimization Constrained optimization Binary optimization Benchmark Swarm intelligence Evolutionary algorithms Particle swarm optimization Genetic algorithm 

1 Introduction

Nature is full of social behaviours for performing different tasks. Although the ultimate goal of all individuals and collective behaviours is survival, creatures cooperate and interact in groups, herds, schools, colonies, and flocks for several reasons: hunting, defending, navigating, and foraging. For instance, Wolf packs own one of the most well-organized social interactions for hunting. Wolves tend to follow a social leadership to hunt preys in different steps: chasing preys, circling preys, harassing preys, and attacking preys [1, 2]. An example of collective defence is schools of fishes in oceans. Thousands of fishes create a school and avoid predators by warning each other, making the predation very difficult for predators [3]. The majority of predators have evolved to divide such schools to sub-schools by attacking them and eventually hunting the separated individuals.

Navigation is another reason for some of the creature to swarm. Birds are the best examples of such behaviours, in which they migrate between continents in flocks conveniently. It has been proven that the v-shaped configuration of flight highly saves the energy and equally distribute drag among the individuals in the flock [4]. Last but not least, foraging is another main reason of social interactions of many species in nature. Ants and bees are the best examples of collective behaviours with the purpose of foraging. It has been proven that ants and bees are able to find and mark the shortest path from the nest/hive to the source of food [5]. They intelligently search for foods and mark the path utilizing pheromone to inform and guide others.

It is very interesting that creatures find the optimal situations and perform tasks efficiently in groups. It is obvious that they have been evolved over centuries to figure out such optimal and efficient behaviours. Therefore, it is quite reasonable that we inspire from them to solve our problems. This is then main purpose of a field of study called swarm intelligence (SI), which was first proposed by Beni and Wang in 1989 [6]. SI refers to the artificial implementation/simulation of the collective and social intelligence of a group of living creatures in nature [7]. Researchers in this field try to figure out the local rules for interactions between the individuals that yield to the social intelligence. Since there is no centralized control unit to guide the individuals, finding the simple rules between some of them can simulate the social behaviour of the whole population.

The ant colony optimization (ACO) algorithm is one of the first SI techniques mimicking the social intelligence of ants when foraging in an ant colony [8, 9]. This algorithm has been inspired from the simple fact that each ant marks its own path towards to food sources outside of the nest by pheromone. Once an ant finds a food source, it goes back to the nest and marks the path by pheromone to show the path to others. When other ants realize such pheromone marks, they also try to follow the path and leave their own pheromones. The key point here is that they might be different paths to the food source. Since a longer path takes longer time to travel for ants, however, the pheromone vaporizes with higher rate before it is re-marked by other ants. Therefore, the shortest path is achieved by simply following the path with stronger level of pheromone and abandoning the paths with weaker pheromone levels. Doringo first inspired from these simple rules and proposed the well-known ACO algorithm [10].

The particle swarm optimization (PSO) algorithm is also another well-regarded SI paradigm. This algorithm mimics the foraging and navigation behaviour of bird flocks and has been proposed by Eberhart and Kennedy [11]. The main inspiration originates from the simple rules of interactions between birds: birds tend to maintain their fly direction towards their current directions, the best location of food source obtained so far, and the best location of the food that the swarm found so far [12]. The PSO algorithm simply mimics these three rules and guides the particles towards the best optimal solutions by each of the individuals and the swarm simultaneously.

The artificial bee colony (ABC) is another recent and popular SI-based algorithm. This algorithm again simulates the social behaviour of honey bees when foraging nectar and has been proposed by Karaboga [13]. The difference of this algorithm compared to ACO and PSO is the division of the honey bees to scout, onlooker, and employed bees [14]. The employed bees are responsible for finding food sources and informing others by a special dance. In addition, onlookers watch the dances, select one of them, and follow the path towards the selected food sources. Scouters discover abandoned food sources and substitute them by new sources.

Since the proposal of these algorithms, a significant number of researchers attempted to improve or apply them in to different problems in diverse fields [15, 16, 17, 18, 19, 20]. The successful application of these algorithms in science and industry evidences the merits of SI-based techniques in practice. The reasons are due to the advantages of SI-based algorithms. Firstly, SI-based techniques save information about the search space over the course of iteration, whereas such information is discarded by evolutionary algorithms (EA) generation by generation. Secondly, there are fewer controlling parameters in SI-based algorithm. Thirdly, SI-based algorithm is equipped with less operators compared to EA algorithms. Finally, SI-based techniques benefit from flexibility, which make them readily applicable to problems in different fields.

Despite the significant number of recent publications in this field [21, 22, 23, 24, 25, 26, 27, 28, 29], there are still other swarming behaviours in nature that have not gained deserved attention. One of the fancy insects that rarely swarm are dragonflies. Since there is no study in the literature to simulate the individual and social intelligence of dragonflies, this paper aims to first find the main characteristics of dragonflies’ swarms. An algorithm is then proposed based on the identified characteristics. The no free lunch (NFL) [30] theorem also supports the motivation of this work to propose this optimizer since this algorithm may outperform other algorithms on some problems that have not been solved so far. The rest of the paper is organized as follows:

Section 2 presents the inspiration and biological foundations of the paper. The mathematical models and the DA algorithm are provided in Sect. 3. This section also proposes binary and multi-objective versions of DA. A comprehensive comparative study on several benchmark functions and one real case study is provided in Sect. 4 to confirm and verify the performances of DA, BDA, and MODA algorithms. Finally, Sect. 5 concludes the work and suggests some directions for future studies.

2 Inspiration

Dragonflies (Odonata) are fancy insects. There are nearly 3000 different species of this insect around the world [31]. As shown in Fig. 1, a dragonfly’s lifecycle includes two main milestones: nymph and adult. They spend the major portion of their lifespan in nymph, and they undergo metamorphism to become adult [31].
Fig. 1

a Real dragonfly, b Life cycle of dragonflies (left image courtesy of Mehrdad Momeny at www.mehrdadmomeny.com)

Dragonflies are considered as small predators that hunt almost all other small insects in nature. Nymph dragonflies also predate on other marine insects and even small fishes. The interesting fact about dragonflies is their unique and rare swarming behaviour. Dragonflies swarm for only two purposes: hunting and migration. The former is called static (feeding) swarm, and the latter is called dynamic (migratory) swarm.

In static swarm, dragonflies make small groups and fly back and forth over a small area to hunt other flying preys such as butterflies and mosquitoes [32]. Local movements and abrupt changes in the flying path are the main characteristics of a static swarm. In dynamic swarms, however, a massive number of dragonflies make the swarm for migrating in one direction over long distances [33].

The main inspiration of the DA algorithm originates from static and dynamic swarming behaviours. These two swarming behaviours are very similar to the two main phases of optimization using meta-heuristics: exploration and exploitation. Dragonflies create sub-swarms and fly over different areas in a static swarm, which is the main objective of the exploration phase. In the static swarm, however, dragonflies fly in bigger swarms and along one direction, which is favourable in the exploitation phase. These two phases are mathematically implemented in the following section.

3 Dragonfly algorithm

3.1 Operators for exploration and exploitation

According to Reynolds, the behaviour of swarms follows three primitive principles [34]:
  • Separation, which refers to the static collision avoidance of the individuals from other individuals in the neighbourhood.

  • Alignment, which indicates velocity matching of individuals to that of other individuals in neighbourhood.

  • Cohesion, which refers to the tendency of individuals towards the centre of the mass of the neighbourhood.

The main objective of any swarm is survival, so all of the individuals should be attracted towards food sources and distracted outward enemies. Considering these two behaviours, there are five main factors in position updating of individuals in swarms as shown in Fig. 2.
Fig. 2

Primitive corrective patterns between individuals in a swarm

Each of these behaviours is mathematically modelled as follows:

The separation is calculated as follows [34]:
$$S_{i} = - \mathop \sum \limits_{j = 1}^{N} X - X_{j}$$
(3.1)
where X is the position of the current individual, Xj shows the position j-th neighbouring individual, and N is the number of neighbouring individuals.
Alignment is calculated as follows:
$$A_{i} = \frac{{\mathop \sum \nolimits_{j = 1}^{N} V_{j} }}{N}$$
(3.2)
where Xj shows the velocity of j-th neighbouring individual.
The cohesion is calculated as follows:
$$C_{i} = \frac{{\mathop \sum \nolimits_{j = 1}^{N} X_{j} }}{N} - X$$
(3.3)
where X is the position of the current individual, N is the number of neighbourhoods, and Xj shows the position j-th neighbouring individual.
Attraction towards a food source is calculated as follows:
$$F_{i} = X^{ + } - X$$
(3.4)
where X is the position of the current individual, and X+ shows the position of the food source.
Distraction outwards an enemy is calculated as follows:
$$E_{i} = X^{ - } + X$$
(3.5)
where X is the position of the current individual, and X shows the position of the enemy.
The behaviour of dragonflies is assumed to be the combination of these five corrective patterns in this paper. To update the position of artificial dragonflies in a search space and simulate their movements, two vectors are considered: step (∆X) and position (X). The step vector is analogous to the velocity vector in PSO, and the DA algorithm is developed based on the framework of the PSO algorithm. The step vector shows the direction of the movement of the dragonflies and defined as follows (note that the position updating model of artificial dragonflies is defined in one dimension, but the introduced method can be extended to higher dimensions):
$$\Delta X_{t + 1} = (sS_{i} + aA_{i} + cC_{i} + fF_{i} + eE_{i} ) + w\Delta X_{t}$$
(3.6)
where s shows the separation weight, Si indicates the separation of the i-th individual, a is the alignment weight, A is the alignment of i-th individual, c indicates the cohesion weight, Ci is the cohesion of the i-th individual, f is the food factor, Fi is the food source of the i-th individual, e is the enemy factor, Ei is the position of enemy of the i-th individual, w is the inertia weight, and t is the iteration counter.
After calculating the step vector, the position vectors are calculated as follows:
$$X_{t + 1} = X_{t} +\Delta X_{t + 1}$$
(3.7)
where t is the current iteration.
With separation, alignment, cohesion, food, and enemy factors (s, a, c, f, and e), different explorative and exploitative behaviours can achieved during optimization. Neighbours of dragonflies are very important, so a neighbourhood (circle in a 2D, sphere in a 3D space, or hyper-sphere in an nD space) with a certain radius is assumed around each artificial dragonfly. An example of swarming behaviour of dragonflies with increasing neighbourhood radius using the proposed mathematical model is illustrated in Fig. 3.
Fig. 3

Swarming behaviour of artificial dragon flies (w = 0.9–0.2, s = 0.1, a = 0.1, c = 0.7, f = 1, e = 1)

As discussed in the previous subsection, dragonflies only show two types of swarms: static and dynamic as shown in Fig. 4. As may be seen in this figure, dragonflies tend to align their flying while maintaining proper separation and cohesion in a dynamic swarm. In a static swarm, however, alignments are very low while cohesion is high to attack preys. Therefore, we assign dragonflies with high alignment and low cohesion weights when exploring the search space and low alignment and high cohesion when exploiting the search space. For transition between exploration and exploitation, the radii of neighbourhoods are increased proportional to the number of iterations. Another way to balance exploration and exploitation is to adaptively tune the swarming factors (s, a, c, f, e, and w) during optimization.
Fig. 4

Dynamic versus static dragonfly swarms

A question may rise here as to how the convergence of dragonflies is guaranteed during optimization. The dragonflies are required to change their weights adaptively for transiting from exploration to exploitation of the search space. It is also assumed that dragonflies tend to see more dragonflies to adjust flying path as optimization process progresses. In other word, the neighbourhood area is increased as well whereby the swarm become one group at the final stage of optimization to converge to the global optimum. The food source and enemy are chosen from the best and worst solutions that the whole swarm is found so far. This causes convergence towards promising areas of the search space and divergence outward non-promising regions of the search space.

To improve the randomness, stochastic behaviour, and exploration of the artificial dragonflies, they are required to fly around the search space using a random walk (Lévy flight) when there is no neighbouring solutions. In this case, the position of dragonflies is updated using the following equation:
$$X_{t + 1} = X_{t} + {\text{L}}{\acute{\text {e}}}{\text{vy}}\left( d \right) \times X_{t}$$
(3.8)
where t is the current iteration, and d is the dimension of the position vectors.
The Lévy flight is calculated as follows [35]:
$${\text{L}}{\acute{\text{e}}}{\text{vy}}\left( x \right) = 0.01 \times \frac{{r_{1} \times \sigma }}{{\left| {r_{2} } \right|^{{\frac{1}{\beta }}} }}$$
(3.9)
where r1, r2 are two random numbers in [0,1], β is a constant (equal to 1.5 in this work), and σ is calculated as follows:
$$\sigma = \left( {\frac{{\varGamma \left( {1 + \beta } \right) \times \sin \left( {\frac{\pi \beta }{2}} \right)}}{{\varGamma \left( {\frac{1 + \beta }{2}} \right) \times \beta \times 2^{{\left( {\frac{\beta - 1}{2}} \right)}} }}} \right)^{1/\beta }$$
(3.10)
where \(\varGamma \left( x \right) = \left( {x - 1} \right)!\).

3.2 The DA algorithm for single-objective problems

The DA algorithm starts optimization process by creating a set of random solutions for a given optimization problems. In fact, the position and step vectors of dragonflies are initialized by random values defined within the lower and upper bounds of the variables. In each iteration, the position and step of each dragonfly are updated using Eqs. (3.7)/(3.8) and (3.6). For updating X and ∆X vectors, neighbourhood of each dragonfly is chosen by calculating the Euclidean distance between all the dragonflies and selecting N of them. The position updating process is continued iteratively until the end criterion is satisfied. The pseudo-codes of the DA algorithm are provided in Fig. 5.
Fig. 5

Pseudo-codes of the DA algorithm

It is worth discussing here that the main differences between the DA and PSO algorithm are the consideration of separation, alignment, cohesion, attraction, distraction, and random walk in this work. Although there are some works in the literature that attempted to integrate separation, alignment, and cohesion to PSO [36, 37, 38], this paper models the swarming behaviour of dragonflies by considering all the possible factors applied to individuals in a swarm. The concepts of static and dynamic swarms are quite novel as well. The proposed model of this work is also completely different from the current improved PSO in the literature cited above.

3.3 The DA algorithm for binary problems (BDA)

Optimization in a binary search space is very different than a continuous space. In continuous search spaces, the search agents of DA are able to update their positions by adding the step vectors to the position vectors. In a binary search space, however, the position of search agents cannot be updated by adding step vectors to X since the position vectors of search agents can only be assigned by 0 or 1. Due to the similarity of DA and other SI techniques, the current methods for solving binary problems in the literature are readily applicable to this algorithm.

According to Mirjalili and Lewis [39], the easiest and most effective method to convert a continuous SI technique to a binary algorithm without modifying the structure is to employ a transfer function. Transfer functions receive velocity (step) values as inputs and return a number in [0,1], which defines the probability of changing positions. The output of such functions is directly proportional to the value of the velocity vector. Therefore, a large value for the velocity of a search agent makes it very likely to update its position. This method simulates abrupt changes in particles with large velocity values similarly to continuous optimization (Fig. 6). Two examples of transfer functions in the literature are illustrated in Fig. 6 [39, 40, 41].
Fig. 6

S-shaped and v-shaped transfer functions

As may be seen in this figure, there are two types of transfer functions: s-shaped versus v-shaped. According to Saremi et al. [40], the v-shaped transfer functions are better than the s-shaped transfer functions because they do not force particles to take values of 0 or 1. In order to solve binary problems with the BDA algorithm, the following transfer function is utilized [39]:
$$T\left( {\Delta x} \right) = \left| {\frac{{\Delta x}}{{\sqrt {\Delta x^{2} + 1} }}} \right|$$
(3.11)
This transfer function is first utilized to calculate the probability of changing position for all artificial dragonflies. The following new updating position formula is then employed to update the position of search agents in binary search spaces:
$$X_{t + 1} = \left\{ {\begin{aligned} {\neg{X_{t}} \quad r < T\left( {\Delta x_{t + 1} } \right)} \\ { X_{t }\quad r \ge T\left( {\Delta x_{t + 1} } \right)} \\ \end{aligned} } \right.$$
(3.12)
where r is a number in the interval of [0,1].
With the transfer function and new position updating equations, the BDA algorithm will be able to solve binary problems easily subject to proper formulation of the problem. It should be noted here that since the distance of dragonflies cannot be determined in a binary space as clearly as a continuous space, the BDA algorithm considers all of the dragonflies as one swarm and simulate exploration/exploitation by adaptively tuning the swarming factors (s, a, c, f, and e) as well as the inertia weigh (w). The pseudo-codes of the BDA algorithm are presented in Fig. 7.
Fig. 7

Pseudo-codes of the BDA algorithm

3.4 The DA algorithm for multi-objective problems (MODA)

Multi-objective problems have multiple objectives, which are mostly in conflict. The answer for such problems is a set of solutions called Pareto optimal solutions set. This set includes Pareto optimal solutions that represent the best trade-offs between the objectives [42]. Without loss of generality, multi-objective optimization can be formulated as a minimization problem as follows:
$${\text{Minimize}}:\;F\left( {\vec{x}} \right) = \left\{ {f_{1} \left( {\vec{x}} \right),\,f_{2} \left( {\vec{x}} \right), \ldots ,\,f_{o} \left( {\vec{x}} \right)} \right\}$$
(3.13)
$${\text{Subject}}\;{\text{to}}: \,g_{i} \left( {\vec{x}} \right) \ge 0, \quad i = 1,2, \ldots ,m$$
(3.14)
$$h_{i} \left( {\vec{x}} \right) = 0,\quad i = 1,2, \ldots ,p$$
(3.15)
$$L_{i} \le x_{i} \le U_{i} , \quad i = 1,2, \ldots ,n$$
(3.16)
where o is the number of objectives, m is the number of inequality constraints, p is the number of equality constraints, and [LiUi] are the boundaries of i-th variable.

Due to the nature of multi-objective problems, the comparison between different solutions cannot be done by arithmetic relational operators. In this case, the concepts of Pareto optimal dominance allow us to compare two solutions in a multi-objective search space. The definitions of Pareto dominance and Pareto optimality are as follows [43]:

Definition 1

Pareto dominance:

Suppose that there are two vectors such as: \(\vec{x} = \left( {x_{1} ,x_{2} , \ldots ,x_{k} } \right)\) and \(\vec{y} = \left( {y_{1} ,y_{2} , \ldots ,y_{k} } \right)\).

Vector x dominates vector y (denote as \(x \succ y\)) iff:
$$\forall i \in \left\{ {1,2, \ldots ,k} \right\}, \left[ {f\left( {x_{i} } \right) \ge f\left( {y_{i} } \right)} \right] \wedge \left[ {\exists i \in 1,2, \ldots ,k:f\left( {x_{i} } \right)} \right]$$
(3.17)

It can be seen in Eq. (3.17) that a solution dominates the other if it shows better or equal values on all objectives (dimensions) and has better value in at last one of the objectives. The definition of Pareto optimality is as follows [44]:

Definition 2

Pareto optimality:

A solution \(\vec{x} \in X\) is called Pareto optimal iff:
$${\nexists } \vec{y} \in X | F\left( {\vec{y}} \right) \succ F\left( {\vec{x}} \right)$$
(3.18)

According to the definition 2, two solutions are non-dominated with respect to each other if neither of them dominates the other. A set including all the non-dominated solutions of a problem is called Pareto optimal set and defined as follows:

Definition 3

Pareto optimal set:

The set of all Pareto optimal solutions is called Pareto set as follows:
$$P_{s} \text{ := }\{ x,y \in X | \exists F(y) \succ F(x)\}$$
(3.19)

A set containing the corresponding objective values of Pareto optimal solutions in Pareto optimal set is called Pareto optimal front. The definition of the Pareto optimal front is as follows:

Definition 4

Pareto optimal front:

A set containing the value of objective functions for Pareto solutions set:
$$P_{f} \text{ := }\{ F(x)|x \in P_{s} \}$$
(3.20)

In order to solve multi-objective problems using meta-heuristics, an archive (repository) is widely used in the literature to maintain the Pareto optimal solutions during optimization. Two key points in finding a proper set of Pareto optimal solutions for a given problem are convergence and coverage. Convergence refers to the ability of a multi-objective algorithm in determining accurate approximations of Pareto optimal solutions. Coverage is the distribution of the obtained Pareto optimal solutions along the objectives. Since most of the current multi-objective algorithms in the literature are posteriori, the coverage and number of solutions are very important for decision making after the optimization process [45]. The ultimate goal for a multi-objective optimizer is to find the most accurate approximation of true Pareto optimal solutions (convergence) with uniform distributions (coverage) across all objectives.

For solving multi-objective problems using the DA algorithm, it is first equipped with an archive to store and retrieve the best approximations of the rue Pareto optimal solutions during optimization. The updating position of search agents is identical to that of DA, but the food sources are selected from the archive. In order to find a well-spread Pareto optimal front, a food source is chosen from the least populated region of the obtained Pareto optimal front, similarly to the multi-objective particle swarm optimization (MOPSO) algorithm [46]. To find the least populated area of the Pareto optimal front, the search space should be segmented. This is done by finding the best and worst objectives of Pareto optimal solutions obtained, defining a hyper-sphere to cover all the solutions, and dividing the hyper-spheres to equal sub-hyper-spheres in each iteration. After the creation of segments, the selection is done by a roulette-wheel mechanism with the following probability for each segment, which was proposed by Coello Coello et al. [47]:
$$P_{i} = \frac{c}{{N_{i} }}$$
(3.21)
where c is a constant number greater than one, and Ni is the number of obtained Pareto optimal solutions in the i-th segment.

This equations allows the MODA algorithm to have higher probability of choosing food sources from the less populated segments. Therefore, the artificial dragonflies will be encouraged to fly around such regions and improve the distribution of the whole Pareto optimal front.

For selecting enemies from the archive, however, the worst (most populated) hyper-sphere should be chosen in order to discourage the artificial dragonflies from searching around non-promising crowded areas. The selection is done by a roulette-wheel mechanism with the following probability for each segment:
$$P_{i} = \frac{{N_{i} }}{c}$$
(3.22)
where c is a constant number greater than one, and Ni is the number of obtained Pareto optimal solutions in the i-th segment.
In may be seen in Eq. (3.22) that the roulette-wheel mechanism assigns high probabilities to the most crowded hyper-spheres for being selected as enemies. An example of the two above-discussed selection processes is illustrated in Fig. 8. Note that the main hyper-sphere that covers all the sub-hyper-spheres is not illustrated in this figure.
Fig. 8

Conceptual model of the best hyper-spheres for selecting a food source or removing a solution from the archive

The archive should be updated regularly in each iteration and may become full during optimization. Therefore, there should be a mechanism to manage the archive. If a solution is dominated by at least one of the archive residences, it should be prevented from entering the archive. If a solution dominates some of the Pareto optimal solutions in the archive, they all should be removed from the archive, and the solution should be allowed to enter the archive. If a solution is non-dominated with respect to all of the solutions in the archive, it should be added to the archive. If the archive is full, one or more than one solutions may be removed from the most populated segments to accommodate new solution(s) in the archive. These rules are taken from the work of Coello Coello et al. [47]. Figure 8 shows the best candidate hyper-sphere (segments) to remove solutions (enemies) from in case the archive become full.

All the parameters of the MODA algorithm are identical to those of the DA algorithm except two new parameters for defining the maximum number of hyper-spheres and archive size. After all, the pseudo-codes of MODA are presented in Fig. 9.
Fig. 9

Pseudo-codes of the MODA algorithm

4 Results and discussion

In this section, a number of test problems and one real case study are selected to benchmark the performance of the proposed DA, BDA, and MODA algorithms.

4.1 Results of DA algorithm

Three groups of test functions with different characteristics are selected to benchmark the performance of the DA algorithm from different perspectives. As shown in Appendix 1, the test functions are divided the three groups: unimodal, multi-modal, and composite functions [48, 49, 50, 51]. As their names imply, unimodal test functions have single optimum, so they can benchmark the exploitation and convergence of an algorithm. In contrast, multi-modal test functions have more than one optimum, which make them more challenging than unimodal functions. One of the optima is called global optimum, and the rest are called local optima. An algorithm should avoid all the local optima to approach and approximate the global optimum. Therefore, exploration and local optima avoidance of algorithms can be benchmarked by multi-modal test functions.

The last group of test functions, composite functions, are mostly the combined, rotated, shifted, and biased version of other unimodal and multi-modal test functions [52, 53]. They mimic the difficulties of real search spaces by providing a massive number of local optima and different shapes for different regions of the search space. An algorithm should properly balance exploration and exploitation to approximate the global optimum of such test functions. Therefore, exploration and exploitation combined can be benchmarked by this group of test functions.

For verification of the results of DA, two well-known algorithms are chosen: PSO [54] as the best algorithm among swarm-based technique and GA [55] as the best evolutionary algorithm. In order to collect quantitative results, each algorithm is run on the test functions 30 times and to calculate the average and standard deviation of the best approximated solution in the last iteration. These two metrics show which algorithm behaves more stable when solving the test functions. Due to the stochastic nature of the algorithms, a statistical test is also conducted to decide about the significance of the results [56]. The averages and standard deviation only compare the overall performance of the algorithms, while a statistical test considers each run’s results and proves that the results are statistically significant. The Wilcoxon nonparametric statistical test [39, 56] is conducted in this work. After all, each of the test functions is solved using 30 search agents over 500 iterations, and the results are presented in Tables 1 and 2. Note that the initial parameters of PSO and GA are identical to the values in the original papers cited above.
Table 1

Statistical results of the algorithms on the test functions

Test function

DA

PSO

GA

Ave

Std

Ave

Std

Ave

Std

TF1

2.85E−18

7.16E−18

4.2E−18

1.31E−17

748.5972

324.9262

TF2

1.49E−05

3.76E−05

0.003154

0.009811

5.971358

1.533102

TF3

1.29E−06

2.1E−06

0.001891

0.003311

1949.003

994.2733

TF4

0.000988

0.002776

0.001748

0.002515

21.16304

2.605406

TF5

7.600558

6.786473

63.45331

80.12726

133307.1

85,007.62

TF6

4.17E−16

1.32E−15

4.36E−17

1.38E−16

563.8889

229.6997

TF7

0.010293

0.004691

0.005973

0.003583

0.166872

0.072571

TF8

−2857.58

383.6466

−7.1E+11

1.2E+12

−3407.25

164.4776

TF9

16.01883

9.479113

10.44724

7.879807

25.51886

6.66936

TF10

0.23103

0.487053

0.280137

0.601817

9.498785

1.271393

TF11

0.193354

0.073495

0.083463

0.035067

7.719959

3.62607

TF12

0.031101

0.098349

8.57E−11

2.71E−10

1858.502

5820.215

TF13

0.002197

0.004633

0.002197

0.004633

68,047.23

87,736.76

TF14

103.742

91.24364

150

135.4006

130.0991

21.32037

TF15

193.0171

80.6332

188.1951

157.2834

116.0554

19.19351

TF16

458.2962

165.3724

263.0948

187.1352

383.9184

36.60532

TF17

596.6629

171.0631

466.5429

180.9493

503.0485

35.79406

TF18

229.9515

184.6095

136.1759

160.0187

118.438

51.00183

TF19

679.588

199.4014

741.6341

206.7296

544.1018

13.30161

Table 2

p values of the Wilcoxon ranksum test over all runs

F

DA

PSO

GA

TF1

N/A

0.045155

0.000183

TF2

N/A

0.121225

0.000183

TF3

N/A

0.003611

0.000183

TF4

N/A

0.307489

0.000183

TF5

N/A

0.10411

0.000183

TF6

0.344704

N/A

0.000183

TF7

0.021134

N/A

0.000183

TF8

0.000183

N/A

0.000183

TF9

0.364166

N/A

0.002202

TF10

N/A

0.472676

0.000183

TF11

0.001008

N/A

0.000183

TF12

0.140465

N/A

0.000183

TF13

N/A

0.79126

0.000183

TF14

N/A

0.909654

0.10411

TF15

0.025748

0.241322

N/A

TF16

0.01133

N/A

0.053903

TF17

0.088973

N/A

0.241322

TF18

0.273036

0.791337

N/A

TF19

N/A

0.472676

N/A

As per the results of the algorithms on the unimodal test functions (TF1–TF7), it is evident that the DA algorithm outperforms PSO and GA on the majority of the cases. The p values in Table 5 also show that this superiority is statistically significant since the p values are less than 0.05. Considering the characteristic of unimodal test functions, it can be stated that the DA algorithm benefits from high exploitation. High exploitation assists the DA algorithm to rapidly converge towards the global optimum and exploit it accurately.

The results of the algorithms on multi-modal test functions (TF8–TF13) show that again the DA algorithm provides very competitive results compared to PSO. The p values reported in Table 2 also show that the DA and PSO algorithms show significantly better results than GA. Considering the characteristics of multi-modal test functions and these results, it may be concluded that the DA algorithm has high exploration which assist it to discover the promising regions of the search space. In addition, the local optima avoidance of this algorithm is satisfactory since it is able to avoid all of the local optima and approximate the global optima on the majority of the multi-modal test functions.

The results of composite test functions (TF14–TF19) show that the DA algorithm provides very competitive results and outperforms others occasionally. However, the p values show that the superiority is not as significant as those of unimodal and multi-modal test functions. This is due to the difficulty of the composite test functions that make them challenging for algorithms employed in this work. Composite test functions benchmark the exploration and exploitation combined. Therefore, these results prove that the operators of the DA algorithm appropriately balance exploration and exploitation to handle difficulty in a challenging search space. Since the composite search spaces are highly similar to the real search spaces, these results make the DA algorithm potentially able to solve challenging optimization problems.

For further observing and analysing the performance of the proposed DA algorithm, four new metrics are employed in the following paragraphs. The main aims of this experiment is to confirm the convergence and predict the potential behaviour of the DA algorithm when solving real problems. The employed quantitative metrics are the position of dragonflies from the first to the last iteration (search history), the value of a parameter from the first to the last iteration (trajectory), the average fitness of dragonflies from the first to the last iteration, and the fitness of the best food source obtained from the first to the last iteration (convergence).

Tracking the position of dragonflies during optimization allows us to observe whether and how the DA algorithm explores and exploits the search space. Monitoring the value of a parameter during optimization assists us to observe the movement of candidate solutions. Preferably, there should be abrupt changes in the parameters in the exploration phase and gradual changes in the exploitation phase. The average fitness of dragonflies during optimization also shows the improvement in the fitness of the whole swarm during optimization. Finally, the fitness of the food source shows the improvement of the obtained global optimum during optimization.

Some of the functions (TF2, TF10, and TF17) are selected and solved by 10 search agents over 150 iterations. The results are illustrated in Figs. 10, 11, 12, and 13. Figure 10 shows the history of dragonfly’s position during optimization. It may be observed that the DA algorithm tends to search the promising regions of the search space extensively. The behaviour of DA when solving TF17, which is a composite test function, is interesting because the coverage of search space seems to be high. This shows that the DA’s artificial dragonflies are able to search the search space effectively.
Fig. 10

Search history of the DA algorithms on unimodal, multi-modal, and composite test functions

Fig. 11

Trajectory of DA’s search agents on unimodal, multi-modal, and composite test functions

Fig. 12

Average fitness of DA’s search agents on unimodal, multi-modal, and composite test functions

Fig. 13

Convergence curve of the DA algorithms on unimodal, multi-modal, and composite test functions

Figure 11 illustrates the trajectory of the first variable of the first artificial dragonfly over 150 iterations. It can be observed that there are abrupt changes in the initial iterations. These changes are decreased gradually over the course of iterations. According to Berg et al. [57], this behaviour can guarantee that an algorithm eventually convergences to a point and search locally in a search space.

Figures 12 and 13 show the average fitness of all dragonflies and the food source, respectively. The average fitness of dragonflies shows a decreasing behaviour on all of the test functions. This proves that the DA algorithm improves the overall fitness of the initial random population. A similar behaviour can be observed in the convergence curves. This also evidences that the approximation of the global optimum becomes more accurate as the iteration counter increases. Another fact that can be seen is the accelerated trend in the convergence curves. This is due to the emphasis on local search and exploitation as iteration increases which highly accelerate the convergence towards the optimum in the final steps of iterations.

As summary, the results of this section proved that the proposed DA algorithm shows high exploration and exploitation. For one, the proposed static swarm promotes exploration, assists the DA algorithm to avoid local optima, and resolves local optima stagnation when solving challenging problems. For another, the dynamic swarm of dragonflies emphasizes exploitation as iteration increases, which causes a very accurate approximation of the global optimum.

4.2 Results of BDA algorithm

To benchmark the performance of the BDA algorithm, test functions TF1 to TF13 are taken from Sect. 4.1 and Appendix 1. For simulating a binary search space, we consider 15 bits to define the variables of the test functions. The dimension of test functions is reduced from 30 to 5, so the total number of binary variables to be optimized by the BDA algorithm is 75 (5 × 15). For verification of the results, the binary PSO (BPSO) [58] and binary gravitational search algorithm (BGSA) [59] are chosen from the literature. Each of the algorithms is run 30 times, and the results are presented in Tables 3 and 4. Note that the initial parameters of BPSO and BGSA are identical to the values in the original papers cited above.
Table 3

Statistical results of the binary algorithms on the test functions

Test function

BDA

BPSO

BGSA

Ave

Std

Ave

Std

Ave

Std

TF1

0.281519

0.417723

5.589032

1.97734

82.95707

49.78105

TF2

0.058887

0.069279

0.196191

0.052809

1.192117

0.228392

TF3

14.23555

22.68806

15.51722

13.68939

455.9297

271.9785

TF4

0.247656

0.330822

1.895313

0.483579

7.366406

2.213344

TF5

23.55335

34.6822

86.44629

65.82514

3100.999

2927.557

TF6

0.095306

0.129678

6.980524

3.849114

106.8896

77.54615

TF7

0.012209

0.014622

0.011745

0.006925

0.03551

0.056549

TF8

−924.481

65.68827

−988.565

16.66224

−860.914

80.56628

TF9

1.805453

1.053829

4.834208

1.549026

10.27209

3.725984

TF10

0.388227

0.5709

2.154889

0.540556

2.786707

1.188036

TF11

0.193437

0.113621

0.47729

0.129354

0.788799

0.251103

TF12

0.149307

0.451741

0.407433

0.231344

9.526426

6.513454

TF13

0.035156

0.056508

0.306925

0.241643

2216.776

5663.491

Table 4

p values of the Wilcoxon ranksum test over all runs

F

BDA

BPSO

BGSA

TF1

N/A

0.000183

0.000183

TF2

N/A

0.001706

0.000183

TF3

N/A

0.121225

0.000246

TF4

N/A

0.000211

0.000183

TF5

N/A

0.009108

0.000183

TF6

N/A

0.000183

0.000183

TF7

0.472676

N/A

0.344704

TF8

0.064022

N/A

0.000583

TF9

N/A

0.000583

0.000183

TF10

N/A

0.00033

0.00044

TF11

N/A

0.000583

0.00033

TF12

N/A

0.002827

0.000183

TF13

N/A

0.000583

0.000183

Table 3 shows that the proposed algorithm outperforms both BPSO and BGSA on the majority of binary test cases. The discrepancy of the results is very evident as per the p values reported in Table 4. These results prove that the BDA algorithm inherits high exploration and exploitation from the DA algorithm due to the use of the v-shaped transfer function.

4.3 Results of MODA algorithm

As multi-objective case studies, five challenging test functions from the well-known ZDT set proposed by Deb et al. [60] are chosen in this subsection. Note that the first three test functions are identical to ZDT1, ZDT2, and ZDT3. However, this paper modifies ZDT1 and ZDT2 to have test problems with linear and tri-objective fronts as the last two case studies. The details of these test functions are available in Appendix 2. The results are collected and discussed quantitatively and qualitatively. Quantitative results are calculated by the inverse generational distance (IGD) proposed by Sierra and Coello Coello [61] over ten runs. This performance metric is similar to generational distance (GD) [62] and formulated as follows:
$${\text{IGD}} = \frac{{\sqrt {\mathop \sum \nolimits_{i = 1}^{n} d_{i}^{2} } }}{n}$$
(4.1)
where n is the number of true Pareto optimal solutions, and di indicates the Euclidean distance between the i-th true Pareto optimal solution and the closest obtained Pareto optimal solutions in the reference set.
For collecting and discussing the qualitative results, the best Pareto optimal front in ten independent runs are presented. The MODA algorithm is compared to MOPSO [47] and non-dominated sorting genetic algorithm (NSGA-II) [63]. After all, the quantitative results are presented in Tables 5, 6, 7, 8, and 9, and the qualitative results are provided in Figs. 14, 15, 16, 17, and 18.
Table 5

Results of the multi-objective algorithms on ZDT1

Algorithm

IGD

Ave

Std

Median

Best

Worst

MODA

0.00612

0.002863

0.0072

0.0024

0.0096

MOPSO

0.00422

0.003103

0.0037

0.0015

0.0101

NSGA-II

0.05988

0.005436

0.0574

0.0546

0.0702

Table 6

Results of the multi-objective algorithms on ZDT2

Algorithm

IGD

Ave

Std

Median

Best

Worst

MODA

0.00398

0.001604244

0.0033

0.0023

0.006

MOPSO

0.00156

0.000174356

0.0017

0.0013

0.0017

NSGA-II

0.13972

0.026263465

0.1258

0.1148

0.1834

Table 7

Results of the multi-objective algorithms on ZDT3

Algorithm

IGD

Ave

Std

Median

Best

Worst

MODA

0.02794

0.004021

0.0302

0.02

0.0304

MOPSO

0.03782

0.006297

0.0362

0.0308

0.0497

NSGA-II

0.04166

0.008073

0.0403

0.0315

0.0557

Table 8

Results of the multi-objective algorithms on ZDT1 with linear front

Algorithm

IGD

Ave

Std

Median

Best

Worst

MODA

0.00616

0.005186

0.0038

0.0022

0.0163

MOPSO

0.00922

0.005531

0.0098

0.0012

0.0165

NSGA-II

0.08274

0.005422

0.0804

0.0773

0.0924

Table 9

Results of the multi-objective algorithms on ZDT2 with three objectives

Algorithm

IGD

Ave

Std

Median

Best

Worst

MODA

0.00916

0.005372

0.0063

0.0048

0.0191

MOPSO

0.02032

0.001278

0.0203

0.0189

0.0225

NSGA-II

0.0626

0.017888

0.0584

0.0371

0.0847

Fig. 14

Best Pareto optimal front obtained by the multi-objective algorithms on ZDT1

Fig. 15

Best Pareto optimal front obtained by the multi-objective algorithms on ZDT2

Fig. 16

Best Pareto optimal front obtained by the multi-objective algorithms on ZDT3

Fig. 17

Best Pareto optimal front obtained by the multi-objective algorithms on ZDT1 with linear front

Fig. 18

Best Pareto optimal front obtained by the multi-objective algorithms on ZDT2 with three objectives

As per the results presented in Tables 5, 6, 7, 8, and 9, the MODA algorithm tends to outperform NSGA-II and provides very competitive results compared to MOPSO on the majority of the test functions. Figures 14, 15, 16, 17, and 18 also show that the convergence and coverage of the Pareto optimal solutions obtained by MODA algorithm are mostly better than NSGA-II. High convergence of the MODA originates from the accelerated convergence of search agents around the food sources selected from the archive over the course of iterations. Adaptive values for s, a, c, f, e, and w in MODA allow its search agents to converge towards the food sources proportional to the number of iterations. High coverage of the MODA algorithm is due to the employed food/enemy selection mechanisms. Since the foods and enemies are selected from the less populated and most populated hyper-spheres, respectively, the search agents of the MODA algorithm tend to search the regions of the search space that have Pareto optimal solutions with low distribution and avoid highly distributed regions in Pareto front. Therefore, the distribution of the Pareto optimal solutions is adjusted and increased along the obtained Pareto optimal front. The maintenance mechanism for a full archive also assists the MODA algorithm to discard excess Pareto optimal solutions (enemies) in populated segments and allows adding new food sources in less populated regions. These results evidence the merits of the proposed MODA in solving multi-objective problems as a posteriori algorithm.

To demonstrate the applicability of the proposed MODA algorithm in practice, a submarine’s propeller is optimized by this algorithm as well. This problem has two objectives: cavitation versus efficiency. These two objectives are in conflict and restricted by a large number of constraints as other computational fluid dynamics (CFD) problems. This problem is formulated as follows:
$${\text{Maximize}}:\eta \left( X \right)$$
(4.2)
$${\text{Miniimize}}:V\left( X \right)$$
(4.3)
$${\text{Subject to}}:T > 40,000, \,{\text{RPM}} = 200,\,Z = 7,\, D = 2, \,d = 0.4,\,{\text{and}}\, S = 5,$$
(4.4)
where η is efficiency, V is cavitation, T is thrust, RPM is rotation per second of the propeller, Z is the number of blades, D is the diameter of the propeller (m), d is the diameter of hub (m), and S is the ship speed (m/s).
The shape of the propeller employed is illustrated in Fig. 19. Note that the full list of constraints and other physical details of the propeller design problem are not provided in this paper, so interested readers are referred to Carlton’s book [64].
Fig. 19

A 7-blade propeller with 2 m diameter for submarines

As shown in Fig. 20, the main structural parameters are the shapes of airfoils along the blades, which define the final shape of the propeller. The structure of each airfoil is determined by two parameters: maximum thickness and chord length. Ten airfoils are considered along the blade in this study, so there is a total of 20 structural parameters to be optimized by the MODA algorithm.
Fig. 20

A blade is divided to ten airfoils each of which has two structural parameters: maximum thickness and chord length

This real case study is solved by the MODA algorithm equipped with 200 artificial dragonflies over 300 iterations. Since the problem of submarine propeller design has many constraints, MODA should be equipped with a constraint-handling method. For simplicity, a death penalty is utilized, which assign very low efficiency and large cavitation to the artificial dragonflies that violate any of the constraints. Therefore, they are dominated automatically when finding non-dominated solutions in the next iteration.

As can be seen in Fig. 21, the MODA algorithm found 61 Pareto optimal solutions for this problem. The low density of searched points (grey dots) is due to the highly constrained nature of this problem. However, it seems that the MODA algorithm successfully improved the initial random designs and determined a very accurate approximation of the true Pareto optimal front. The solutions are highly distributed along both objectives, which confirm the coverage of this algorithm in practice as well. Therefore, these results prove the convergence and coverage of the MODA algorithm in solving real problems with unknown true Pareto optimal front. Since the propeller design problem is highly constrained, these results also evidence the merits of the proposed MODA algorithm in solving challenging constrained problems as well.
Fig. 21

Search history, obtained Pareto optimal front, and shape of some of the obtained Pareto optimal solutions by MODA

5 Conclusion

This paper proposed another SI algorithm inspired by the behaviour of dragonflies’ swarms in nature. Static and dynamic swarming behaviours of dragonflies were implemented to explore and exploit the search space, respectively. The algorithm was equipped with five parameters to control cohesion, alignment, separation, attraction (towards food sources), and distraction (outwards enemies) of individuals in the swarm. Suitable operators were integrated to the proposed DA algorithm for solving binary and multi-objective problems as well. A series of continuous, binary, and multi-objective test problems were employed to benchmark the performance of the DA, BDA, and MODA algorithms from different perspectives. The results proved that all of the proposed algorithms benefits from high exploration, which is due to the proposed static swarming behaviour of dragonflies. The convergence of the artificial dragonflies towards optimal solutions in continuous, binary, and multi-objective search spaces was also observed and confirmed, which are due to the dynamic swarming pattern modelled in this paper.

The paper also considered designing a real propeller for submarines using the proposed MODA algorithm, which is a challenging and highly constrained CFD problem. The results proved the effectiveness of the multi-objective version of DA in solving real problems with unknown search spaces. As per the finding of this comprehensive study, it can be concluded that the proposed algorithms are able to outperform the current well-known and powerful algorithms in the literature. Therefore, they are recommended to researchers from different fields as open-source optimization tools. The source codes of DA, BDA, and MODA are publicly available at http://www.alimirjalili.com/DA.html.

For future works, several research directions can be recommended. Hybridizing other algorithms with DA and integrating evolutionary operators to this algorithm are two possible research avenues. For the BDA algorithm, the effects of transfer functions on the performance of this algorithm worth to be investigated. Applying other multi-objective optimization approaches (non-dominated sorting for instance) to MODA will also be valuable contributions. The DA, BDA, and MODA algorithm can all be tuned and employed to solve optimization problems in different fields as well.

Notes

Acknowledgments

The author would like to thank Mehrdad Momeny for providing his outstanding dragonfly photo.

Supplementary material

521_2015_1920_MOESM1_ESM.gif (460 kb)
Supplementary material 1 (GIF 460 KB)

References

  1. 1.
    Mirjalili S, Mirjalili SM, Lewis A (2014) Grey wolf optimizer. Adv Eng Softw 69:46–61CrossRefGoogle Scholar
  2. 2.
    Muro C, Escobedo R, Spector L, Coppinger R (2011) Wolf-pack (Canis lupus) hunting strategies emerge from simple rules in computational simulations. Behav Process 88:192–197CrossRefGoogle Scholar
  3. 3.
    Jakobsen PJ, Birkeland K, Johnsen GH (1994) Swarm location in zooplankton as an anti-predator defence mechanism. Anim Behav 47:175–178CrossRefGoogle Scholar
  4. 4.
    Higdon J, Corrsin S (1978) Induced drag of a bird flock. Am Nat 112(986):727–744Google Scholar
  5. 5.
    Goss S, Aron S, Deneubourg J-L, Pasteels JM (1989) Self-organized shortcuts in the Argentine ant. Naturwissenschaften 76:579–581CrossRefGoogle Scholar
  6. 6.
    Beni G, Wang J (1993) Swarm intelligence in cellular robotic systems. In: Dario P, Sandini G, Aebischer P (eds) Robots and biological systems: towards a new bionics? NATO ASI series, vol 102. Springer, Berlin, Heidelberg, pp 703–712Google Scholar
  7. 7.
    Bonabeau E, Dorigo M, Theraulaz G (1999) Swarm intelligence: from natural to artificial systems. Oxford University Press, OxfordMATHGoogle Scholar
  8. 8.
    Dorigo M, Stützle T (2003) The ant colony optimization metaheuristic: algorithms, applications, and advances. In: Glover F, Kochenberger GA (eds) Handbook of metaheuristics. International series in operations research & management science, vol 57. Springer, USA, pp 250–285Google Scholar
  9. 9.
    Dorigo M, Maniezzo V, Colorni A (1996) Ant system: optimization by a colony of cooperating agents. Syst Man Cybern Part B Cybern IEEE Trans 26:29–41CrossRefGoogle Scholar
  10. 10.
    Colorni A, Dorigo M, Maniezzo V (1991) Distributed optimization by ant colonies. In: Proceedings of the first European conference on artificial life, pp 134–142Google Scholar
  11. 11.
    Eberhart RC, Kennedy J (1995) A new optimizer using particle swarm theory. In: Proceedings of the sixth international symposium on micro machine and human science, pp 39–43Google Scholar
  12. 12.
    Eberhart RC, Shi Y (2001) Particle swarm optimization: developments, applications and resources. In: Proceedings of the 2001 congress on evolutionary computation, pp 81–86Google Scholar
  13. 13.
    Karaboga D (2005) An idea based on honey bee swarm for numerical optimization. In: Technical report-tr06, Erciyes university, engineering faculty, computer engineering departmentGoogle Scholar
  14. 14.
    Karaboga D, Basturk B (2007) A powerful and efficient algorithm for numerical function optimization: artificial bee colony (ABC) algorithm. J Global Optim 39:459–471MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    AlRashidi MR, El-Hawary ME (2009) A survey of particle swarm optimization applications in electric power systems. Evolut Comput IEEE Trans 13:913–918CrossRefGoogle Scholar
  16. 16.
    Wei Y, Qiqiang L (2004) Survey on particle swarm optimization algorithm. Eng Sci 5:87–94Google Scholar
  17. 17.
    Chandra Mohan B, Baskaran R (2012) A survey: ant colony optimization based recent research and implementation on several engineering domain. Expert Syst Appl 39:4618–4627CrossRefGoogle Scholar
  18. 18.
    Dorigo M, Stützle T (2010) Ant colony optimization: overview and recent advances. In: Gendreau M, Potvin J-Y (eds) Handbook of metaheuristics. International series in operations research & management science, vol 146. Springer, USA, pp 227–263Google Scholar
  19. 19.
    Karaboga D, Gorkemli B, Ozturk C, Karaboga N (2014) A comprehensive survey: artificial bee colony (ABC) algorithm and applications. Artif Intell Rev 42:21–57CrossRefGoogle Scholar
  20. 20.
    Sonmez M (2011) Artificial Bee Colony algorithm for optimization of truss structures. Appl Soft Comput 11:2406–2418CrossRefGoogle Scholar
  21. 21.
    Wang G, Guo L, Wang H, Duan H, Liu L, Li J (2014) Incorporating mutation scheme into krill herd algorithm for global numerical optimization. Neural Comput Appl 24:853–871CrossRefGoogle Scholar
  22. 22.
    Wang G-G, Gandomi AH, Alavi AH (2014) Stud krill herd algorithm. Neurocomputing 128:363–370CrossRefGoogle Scholar
  23. 23.
    Wang G-G, Gandomi AH, Alavi AH (2014) An effective krill herd algorithm with migration operator in biogeography-based optimization. Appl Math Model 38:2454–2462MathSciNetCrossRefGoogle Scholar
  24. 24.
    Wang G-G, Gandomi AH, Alavi AH, Hao G-S (2014) Hybrid krill herd algorithm with differential evolution for global numerical optimization. Neural Comput Appl 25:297–308CrossRefGoogle Scholar
  25. 25.
    Wang G-G, Gandomi AH, Zhao X, Chu HCE (2014) Hybridizing harmony search algorithm with cuckoo search for global numerical optimization. Soft Comput. doi:10.1007/s00500-014-1502-7
  26. 26.
    Wang G-G, Guo L, Gandomi AH, Hao G-S, Wang H (2014) Chaotic krill herd algorithm. Inf Sci 274:17–34MathSciNetCrossRefGoogle Scholar
  27. 27.
    Wang G-G, Lu M, Dong Y-Q, Zhao X-J (2015) Self-adaptive extreme learning machine. Neural Comput Appl. doi:10.1007/s00521-015-1874-3
  28. 28.
    Mirjalili S (2015) The ant lion optimizer. Adv Eng Softw 83:80–98Google Scholar
  29. 29.
    Mirjalili S, Mirjalili SM, Hatamlou A (2015) Multi-Verse Optimizer: a nature-inspired algorithm for global optimization. Neural Comput Appl. doi:10.1007/s00521-015-1870-7
  30. 30.
    Wolpert DH, Macready WG (1997) No free lunch theorems for optimization. Evolut Comput IEEE Trans 1(1):67–82CrossRefGoogle Scholar
  31. 31.
    Thorp JH, Rogers DC (2014) Thorp and Covich’s freshwater invertebrates: ecology and general biology. Elsevier, AmsterdamGoogle Scholar
  32. 32.
    Wikelski M, Moskowitz D, Adelman JS, Cochran J, Wilcove DS, May ML (2006) Simple rules guide dragonfly migration. Biol Lett 2:325–329CrossRefGoogle Scholar
  33. 33.
    Russell RW, May ML, Soltesz KL, Fitzpatrick JW (1998) Massive swarm migrations of dragonflies (Odonata) in eastern North America. Am Midl Nat 140:325–342CrossRefGoogle Scholar
  34. 34.
    Reynolds CW (1987) Flocks, herds and schools: a distributed behavioral model. ACM SIGGRAPH Comput Gr 21:25–34CrossRefGoogle Scholar
  35. 35.
    Yang X-S (2010) Nature-inspired metaheuristic algorithms, 2nd edn. Luniver PressGoogle Scholar
  36. 36.
    Cui Z, Shi Z (2009) Boid particle swarm optimisation. Int J Innov Comput Appl 2:77–85CrossRefGoogle Scholar
  37. 37.
    Kadrovach BA, Lamont GB (2002) A particle swarm model for swarm-based networked sensor systems. In: Proceedings of the 2002 ACM symposium on applied computing, pp 918–924Google Scholar
  38. 38.
    Cui Z (2009) Alignment particle swarm optimization. In: Cognitive informatics, 2009. ICCI’09. 8th IEEE international conference on, pp 497–501Google Scholar
  39. 39.
    Mirjalili S, Lewis A (2013) S-shaped versus V-shaped transfer functions for binary particle swarm optimization. Swarm Evolut Comput 9:1–14CrossRefGoogle Scholar
  40. 40.
    Saremi S, Mirjalili S, Lewis A (2014) How important is a transfer function in discrete heuristic algorithms. Neural Comput Appl:1–16Google Scholar
  41. 41.
    Mirjalili S, Wang G-G, Coelho LDS (2014) Binary optimization using hybrid particle swarm optimization and gravitational search algorithm. Neural Comput Appl 25:1423–1435CrossRefGoogle Scholar
  42. 42.
    Mirjalili S, Lewis A (2015) Novel performance metrics for robust multi-objective optimization algorithms. Swarm Evolut Comput 21:1–23CrossRefGoogle Scholar
  43. 43.
    Coello CAC (2009) Evolutionary multi-objective optimization: some current research trends and topics that remain to be explored. Front Comput Sci China 3:18–30CrossRefGoogle Scholar
  44. 44.
    Ngatchou P, Zarei A, El-Sharkawi M (2005) Pareto multi objective optimization. In: Intelligent systems application to power systems, 2005. Proceedings of the 13th international conference on, pp 84–91Google Scholar
  45. 45.
    Branke J, Kaußler T, Schmeck H (2001) Guidance in evolutionary multi-objective optimization. Adv Eng Softw 32:499–507CrossRefMATHGoogle Scholar
  46. 46.
    Coello Coello CA, Lechuga MS (2002) MOPSO: A proposal for multiple objective particle swarm optimization. In: Evolutionary computation, 2002. CEC’02. Proceedings of the 2002 congress on, pp 1051–1056Google Scholar
  47. 47.
    Coello CAC, Pulido GT, Lechuga MS (2004) Handling multiple objectives with particle swarm optimization. Evolut Comput IEEE Trans 8:256–279CrossRefGoogle Scholar
  48. 48.
    Yao X, Liu Y, Lin G (1999) Evolutionary programming made faster. Evolut Comput IEEE Trans 3:82–102CrossRefGoogle Scholar
  49. 49.
    Digalakis J, Margaritis K (2001) On benchmarking functions for genetic algorithms. Int J Comput Mathematics 77:481–506MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Molga M, Smutnicki C (2005) Test functions for optimization needs. Test functions for optimization needs. http://www.robertmarks.org/Classes/ENGR5358/Papers/functions.pdf
  51. 51.
    Yang X-S (2010) Test problems in optimization. arXiv preprint arXiv:1008.0549
  52. 52.
    Liang J, Suganthan P, Deb K (2005) Novel composition test functions for numerical global optimization. In: Swarm intelligence symposium, 2005. SIS 2005. Proceedings 2005 IEEE, pp 68–75Google Scholar
  53. 53.
    Suganthan PN, Hansen N, Liang JJ, Deb K, Chen Y, Auger A et al (2005) Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. In: KanGAL Report, vol 2005005Google Scholar
  54. 54.
    Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Neural networks, 1995. Proceedings, IEEE International conference on, pp 1942–1948Google Scholar
  55. 55.
    John H (1992) Holland, adaptation in natural and artificial systems. MIT Press, CambridgeGoogle Scholar
  56. 56.
    Derrac J, García S, Molina D, Herrera F (2011) A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms. Swarm Evolut Comput 1:3–18CrossRefGoogle Scholar
  57. 57.
    van den Bergh F, Engelbrecht A (2006) A study of particle swarm optimization particle trajectories. Inf Sci 176:937–971MathSciNetCrossRefMATHGoogle Scholar
  58. 58.
    J. Kennedy J, Eberhart RC (1997) A discrete binary version of the particle swarm algorithm. In: Systems, man, and cybernetics, 1997. computational cybernetics and simulation, 1997 IEEE international conference on, pp 4104–4108Google Scholar
  59. 59.
    Rashedi E, Nezamabadi-Pour H, Saryazdi S (2010) BGSA: binary gravitational search algorithm. Nat Comput 9:727–745MathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    Zitzler E, Deb K, Thiele L (2000) Comparison of multiobjective evolutionary algorithms: empirical results. Evol Comput 8:173–195CrossRefGoogle Scholar
  61. 61.
    Sierra MR, Coello Coello CA (2005) Improving PSO-based multi-objective optimization using crowding, mutation and ∈-dominance. In: Coello Coello CA, Hernández Aguirre A, Zitzler E (eds) Evolutionary multi-criterion optimization. Lecture notes in computer science, vol 3410. Springer, Berlin, Heidelberg, pp 505–519Google Scholar
  62. 62.
    Van Veldhuizen DA, Lamont GB (1998) Multiobjective evolutionary algorithm research: a history and analysis (Final Draft) TR-98-03Google Scholar
  63. 63.
    Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. Evolut Comput IEEE Trans 6:182–197CrossRefGoogle Scholar
  64. 64.
    Carlton J (2012) Marine propellers and propulsion. Butterworth-Heinemann, OxfordGoogle Scholar

Copyright information

© The Natural Computing Applications Forum 2015

Authors and Affiliations

  1. 1.School of Information and Communication TechnologyGriffith UniversityBrisbaneAustralia
  2. 2.Queensland Institute of Business and TechnologyMt Gravatt, BrisbaneAustralia

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